Regen
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on in
Ge
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i
Mayy 2014
777777
THE STATE EDUCATION DEPARTMENT / THE UNIVERSITY OF THE STATE OF NEW YORK / ALBANY, NY 12234
New York State Common Core Sample Questions:
Regents Examination in Geometry (Common Core)
With the adoption of the New York P-12 Common Core Learning Standards (CCLS) in ELA/Literacy
and Mathematics, the Board of Regents signaled a shift in both instruction and assessment.
Educators around the state have already begun instituting Common Core instruction in their
classrooms. To aid in this transition, we are providing sample Regents Examination in Geometry
(Common Core) questions to help students, parents, and educators better understand the
instructional shifts demanded by the Common Core and the rigor required to ensure that all
students are on track to college and career readiness.
These Questions Are Teaching Tools
The sample questions emphasize the instructional shifts demanded by the Common Core. For
Geometry (Common Core) we have provided fourteen questions. These questions include
multiple-choice and constructed response. The sample questions are teaching tools for educators
and can be shared freely with students and parents. They are designed to help clarify the way the
Common Core should drive instruction and how students will be assessed on the Geometry
Regents Examination in Geometry measuring CCLS beginning in June 2015. NYSED is eager for
feedback on these sample questions. Your input will guide us as we develop future exams.
These Questions Are NOT Test Samplers
While educators from around the state have helped craft these sample questions, they have not
undergone the same extensive review, vetting, and field testing that occurs with actual questions
used on the State exams. The sample questions were designed to help educators think about
content, NOT to show how operational exams look exactly or to provide information about how
teachers should administer the test.
How to Use the Sample Questions




Interpret how the standards are conceptualized in each question.
Note the multiple ways the standards are assessed throughout the sample questions.
Look for opportunities for mathematical modeling, i.e., connecting mathematics with the
real world by conceptualizing, analyzing, interpreting, and validating conclusions in order to
make decisions about situations in everyday life, society, or the workplace.
Consider the instructional changes that will need to occur in your classroom.
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May 2014




Notice the application of mathematical ways of thinking to real-world issues and
challenges.
Pay attention to the strong distractors in each multiple-choice question.
Don’t consider these questions to be the only way the standards will be assessed.
Don’t assume that the sample questions represent a mini-version of future State exams.
Understanding Math Sample Questions
Multiple-Choice Questions
Sample multiple-choice math questions are designed to assess CCLS math standards. Math
multiple-choice questions assess procedural fluency and conceptual understanding. Unlike
questions on past math exams, many require the use of multiple skills and concepts. Within the
sample questions, all distractors will be based on plausible missteps.
Constructed Response Questions
Math constructed response questions are similar to past questions, asking students to show their
work in completing one or more tasks or more extensive problems. Constructed response
questions allow students to show their understanding of math procedures, conceptual
understanding, and application.
Format of the Math Sample Questions Document
The Math Sample Questions document is formatted so that headings appear below each item to
provide information for teacher use to help interpret the item, understand measurement with the
CCLS, and inform instruction. A list of the headings with a brief description of the associated
information is shown below.
Key: This is the correct response or, in the case of multiple-choice items, the correct option.
Measures CCLS: This item measures the knowledge, skills, and proficiencies characterized by the
standards within the identified cluster.
Mathematical Practices: If applicable, this is a list of mathematical practices associated with the
item.
Commentary: This is an explanation of how the item measures the knowledge, skills, and
proficiencies characterized by the identified cluster.
Rationale: For multiple-choice items, this section provides the correct option and demonstrates
one method for arriving at that response. For constructed response items, one
possible approach to solving the item is shown followed by the scoring rubric that is
specific to the item. Note that there are often multiple approaches to solving each
problem. The rationale section provides only one example. The scoring rubrics should
be used to evaluate the efficacy of different methods of arriving at a solution.
iii
May 2014
Common Core Sample Question #1
1
What are the coordinates of the point on the directed line segment from K(–5,–4) to
L(5,1) that partitions the segment into a ratio of 3 to 2?
(1) (–3,–3)
(2) (–1,–2)
3
(3) (0,  )
2
(4) (1,–1)
1
May 2014
Key: 4
Measures CCLS: G-GPE.B
Mathematical Practice: 2, 7
Commentary: This question measures G-GPE.B because the student needs to find the
coordinates of a point dividing a directed line segment into the ratio of 3
to 2.
Rationale: Option 4 is correct. Since KL is a directed line segment, the point dividing
KL into a ratio of 3 to 2 is
3
the distance from point ‫ ܭ‬to point ‫ܮ‬.
5
x
y
5  (5   5)
4  (1   4)
5  (10)
3
4  (5)
5 + 6
4 + 3
1
1
3
5
3
5
3
5
5
(1, 1 )
2
May 2014
Common Core Sample Question #2
2
A regular pentagon is shown in the diagram below.
If the pentagon is rotated clockwise around its center, the minimum number of
degrees it must be rotated to carry the pentagon onto itself is
(1)
(2)
(3)
(4)
54º
72º
108º
360º
3
May 2014
Key: 2
Measures CCLS: G-CO.A
Mathematical Practice: 2, 7
Commentary: This question measures G-CO.A because it requires the student to
describe a rotation that carries a regular pentagon onto itself.
Rationale: Option 2 is correct. Segments drawn from the center of the regular pentagon
bisect each angle of the pentagon, and create five isosceles triangles as shown
in the diagram below. Since each exterior angle equals the angles formed by
the segments drawn from the center of the regular pentagon, the minimum
degrees necessary to carry a regular polygon onto itself are equal to the
measure of an exterior angle of the regular polygon.
360
5
4
 72.
May 2014
Common Core Sample Question #3
3
The equation of line h is 2 x  y  1. Line m is the image of line h after a dilation of
scale factor 4 with respect to the origin. What is the equation of the line m?
(1)
(2)
(3)
(4)
y = –2x + 1
y = –2x + 4
y = 2x + 4
y = 2x + 1
5
May 2014
Key: 2
Meaures CCLS: G-SRT.A
Mathematical Practice: 2
Commentary: This question measures G-SRT.A because a line that is dilated and does
not pass through the center of dilation results in a parallel line.
Rationale: Option 2 is correct. The given line h, 2x + y = 1, does not pass through the
center of dilation, the origin, because the y-intercept is at (0,1). The slope of
the dilated line, m, will remain the same as the slope of line h, –2. All points
on line h, such as (0,1), the y-intercept, are dilated by a scale factor of 4;
therefore, the y-intercept of the dilated line is (0,4) because the center of
dilation is the origin, resulting in the dilated line represented by the equation
y = –2x + 4.
6
May 2014
Common Core Sample Question #4
4
As shown in the diagram below, circle A as a radius of 3 and circle B has a radius of
5.
Use transformations to explain why circles A and B are similar.
7
May 2014
Key: See explanation in the rationale below. A correct explanation must include a written
verbal statement.
Measures CCLS: G-C.A
Mathematical Practice: 3, 6
Commentary: This question measures G-C.A because the student must explain why two
given circles are similar.
Rationale: Circle
 ‫ ܣ‬can be mapped onto circle ‫ ܤ‬by first translating circle ‫ ܣ‬along vector
AB such that ‫ ܣ‬maps onto B, and then dilating circle A, centered at A, by a
scale factor of
5
3
. Since there exists a sequence of transformations that maps
circle ‫ ܣ‬onto circle ‫ܤ‬, circle ‫ ܣ‬is similar to circle ‫ܤ‬.
Rubric:
[2]
A complete and correct explanation is written.
[1]
An appropriate explanation is written, but one computational error is made.
or
An appropriate explanation is written, but one conceptual error is made.
[1]
[0]
A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
8
May 2014
Common Core Sample Question #5
5
Two stacks of 23 quarters each are shown below. One stack forms a cylinder but the
other stack does not form a cylinder.
Use Cavelieri’s principle to explain why the volumes of these two stacks of quarters
are equal.
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May 2014
Key: See explanation in rationale below.
Measures CCLS: G-GMD.A
Mathematical Practice: 3, 6
Commentary: This question measures G-GMD.A because the student is required to
explain the relationship of the volumes of two objects using Cavelieri’s
principle.
Rationale: Each quarter in both stacks has the same base area. Therefore, each
corresponding cross-section of the stacks will have the same area. Since the
two stacks of quarters have the same height of 23 quarters, the two volumes
must be the same.
Rubric:
[2]
A complete and correct explanation is written.
[1]
An appropriate explanation is written, but one conceptual error is made.
or
An incomplete or partially correct explanation is written.
[1]
[0]
A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
10
May 2014
Common Core Sample Question #6
6
In the diagram below, triangles XYZ and UVZ are drawn such that X  U and
XZY  UZV.
Describe a sequence of similarity transformations that shows
UVZ .
11
XYZ
is similar to
May 2014
Key: See the description in the rationale below.
Measures CCLS: G-SRT.A
Mathematical Practice: 3, 6
Commentary: This question measures G-SRT.A because students must describe a
sequence of similarity transformations to show two triangles are similar
when they have two pairs of corresponding angles congruent.
Rationale: Triangle X Y Z  is the image of XYZ after a rotation about point ܼ such
that ZX coincides with ZU . Since rotations preserve angle measure, ZY
coincides with ZV, and corresponding angles X and Y, after the rotation,
remain congruent, so XY  UV .
Then, dilate
X Y Z  by a scale factor of
ZU
ZX
with its center at point Z.
Since dilations preserve parallelism, XY maps onto UV . Therefore,
XYZ  UVZ .
Rubric:
[2]
A correct description of sequence of similarity transformations is written.
[1]
One conceptual error is made, but an appropriate sequence of similarity
transformations is written.
or
An incomplete or partially correct sequence of similarity transformations is written.
[1]
[0]
A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
12
May 2014
Common Core Sample Question #7
7
Explain why cos(x) = sin(90 – x) for x such that 0 < x < 90.
13
May 2014
Key: See explanation in the rationale below. A correct explanation must include a written
verbal statement.
Measures CCLS: G-SRT.C
Mathematical Practice: 3, 6
Commentary: This question measures G-SRT.C because the student is required to
explain why the sine and cosine of complementary angles are equal.
Rationale: The acute angles in a right triangle are always complementary. The sine of
any acute angle is equal to the cosine of its complement.
Rubric:
[2]
A correct and complete explanation is written.
[1]
One conceptual error is made, but an appropriate explanation is written.
or
[1] An incomplete or partially correct explanation is written.
[0]
A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
14
May 2014
Common Core Sample Question #8
8
In the diagram of
LAC and DNC
below, LA  DN , CA  CN , and DAC  LCN .
LAC  DNC.
a)
Prove that
b)
Describe a sequence of rigid motions that will map
15
LAC onto DNC.
May 2014
Key: See rationale below. A sequence of one transformation is acceptable.
Measures CCLS: G-SRT.B, G-CO.A
Mathematical Practice: 3, 6
Commentary: This question measures G-SRT.B and G-CO.A because students are
required to prove two triangles are congruent and demonstrate
congruence using rigid motion.
Rationale:
Statement
Reasons
1. LA  DN , CA  CN , and DC  LN
1. Given
2. LCA and DCN are right angles
2. Definition of perpendicular lines
3.
4.
LAC and DNC
LAC  DNC
3. Definition of a right triangle
are right triangles
4. H.L. Theorem
Triangle LAC will map onto DNC after rotating LAC counterclockwise 90 degrees
about point C such that point ‫ ܮ‬maps onto point D.
Rubric:
Part a)
[2]
A complete and correct proof that includes a conclusion is written.
[1]
Only one correct statement and reason are written.
or
[1]
One conceptual error is made.
[0]
The “given” and/or the “prove” statements are written, but no further correct
relevant statements are written.
or
A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
[0]
Part b)
LAC onto DNC
is written.
[2]
A complete and correct description mapping
[1]
An appropriate description is written, but one conceptual error is made.
[0]
A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
16
May 2014
Common Core Sample Question #9
9
As shown below, a canoe is approaching a lighthouse on the coastline of a lake. The
front of the canoe is 1.5 feet above the water and an observer in the lighthouse is 112
feet above the water.
At 5:00, the observer in the lighthouse measured the angle of depression to the front
of the canoe to be 6°. Five minutes later, the observer measured and saw the angle of
depression to the front of the canoe had increased by 49°. Determine and state, to the
nearest foot per minute, the average speed at which the canoe traveled toward the
lighthouse.
17
May 2014
Key: 195
Measures CCLS: G-SRT.C
Mathematical Practice: 1, 4
Commentary: This question measures G-SRT.C because students need to use modeling
and trigonometric ratios to find the average speed.
Rationale: x represents the distance between the lighthouse and the canoe at 5:00.
y represents the distance between the lighthouse and the canoe at 5:05.
tan 6 =
x=
112  1.5
x
tan (6 + 49) =
110.5
y=
tan 6
x = 1051.337272
Average speed =
112  1.5
y
110.5
tan 55
y = 77.37293297
973.964339 ft
5 min
 194.7928678
Average speed = 195 ft/min
Rubric:
[4]
195, and correct work is shown.
[3]
Appropriate work is shown, but one computational or rounding error is made.
or
Appropriate work is shown to find the distance traveled, but no further correct work
is shown.
[3]
[2]
[2]
[2]
[1]
Appropriate work is shown, but two computational or rounding errors are made.
or
Appropriate work is shown, but one conceptual error is made.
or
Appropriate work is shown to find the distance from the lighthouse at 5:00 and at
5:05, but no further correct work is shown.
Appropriate work is shown, but one computational or rounding error and one
conceptual error are made.
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May 2014
[1]
[1]
[0]
or
Appropriate work is shown to find the distance from the lighthouse at either 5:00 or
at 5:05, but no further correct work is shown.
or
195, but no work is shown.
A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
19
May 2014
Common Core Sample Question #10
10 In the diagram below of circle O, diameter AB and radii OC and OD are drawn.
The length of AB is 12 and the measure of COD is 20 degrees.
 , find the area of sector BOD in terms of π.
AC  BD
If 
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May 2014
Key: 8π
Measures CCLS: G-C.B
Mathematical Practice: 2
Commentary: This question measures G-C.B because students are required to find the
area of a sector.
Rationale: Ao represents the area of circle O and As represents the area of sector BOD.
Ao = πr 2
Ao = π(6)
mBOD 
2
180  20
2
mBOD  80
Ao = 36π
As
36

80
360
360( As )  2880π
As = 8π
Rubric:
[4]
8π, and correct work is shown.
[3]
Appropriate work is shown, but one computational error is made.
or
Appropriate work is shown, but the area of the sector is written as an appropriate
decimal.
[3]
[2]
[2]
[2]
[1]
[1]
[1]
[0]
Appropriate work is shown, but two computational errors are made.
or
Appropriate work is shown, but one conceptual error is made.
or
Appropriate work is shown to find 36 the area of the circle, and 80, the measure
of angle BOD, but no further correct work is shown.
Appropriate work is shown, but one computational error and one conceptual error
are made.
or
Appropriate work is shown to find either 36 the area of the circle, or 80, the
measure of angle BOD, but no further correct work is shown.
or
8π, but no work is shown.
A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
21
May 2014
Common Core Sample Question #11
11 Given:
XYZ ,
XY  ZY , and YW bisects XYZ
Prove that ∠YWZ is a right angle.
22
May 2014
Key: See proof in the rationale below.
Measures CCLS: G-CO.C
Mathematical Practice: 3, 6
Commentary: This question measures G-CO.C because students are required to prove
that the altitude of an isosceles triangle forms right angles.
Rationale: Multiple methods of proof are acceptable.
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May 2014
or
Statements
1.
Reasons
XYZ , XY  ZY , YW
bisects
1. Given
XYZ
2.
XYZ
2. Definition of isosceles triangle
is isosceles
3. YW is an altitude of
XYZ
3. The angle bisector of the vertex of an
isosceles triangle is also the altitude of that
triangle.
4. YW  XZ
4. Definition of altitude
5. YWZ is a right angle
5. Definition of perpendicular lines
Rubric:
[4]
A complete and correct proof that includes a concluding statement is written.
[3]
A proof is written that demonstrates a thorough understanding of the method of
proof and contains no conceptual errors, but one statement and/or reason is missing
or is incorrect, or the concluding statement is missing.
[2]
A proof is written that demonstrates a good understanding of the method of proof
and contains no conceptual errors, but two statements and/or reasons are missing or
are incorrect.
or
A proof is written that demonstrates a good understanding of the method of proof,
but one conceptual error is made.
[2]
[1]
Only one correct relevant statement and reason are written.
[0]
The “given” and/or the “prove” statements are written, but no further correct
relevant statements are written.
or
A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
[0]
24
May 2014
Common Core Sample Question #12
12 Trees that are cut down and stripped of their branches for timber are approximately
cylindrical. A timber company specializes in a certain type of tree that has a typical
diameter of 50 cm and a typical height of about 10 meters. The density of the wood
is 380 kilograms per cubic meter, and the wood can be sold by mass at a rate of
$4.75 per kilogram. Determine and state the minimum number of whole trees that
must be sold to raise at least $50,000.
25
May 2014
Key: 15
Measures CCLS: G-MG.A
Mathematical Practice: 1, 4
Commentary: This question measures G-MG.A because a cylinder is used to model a
tree trunk to solve the problem. This problem requires students to
navigate multiple steps and develop an appropriate model.
Rationale:
Volume of one tree
Weight of one tree (x)
380 K
V = πr 2 h
1m
V = π(0.25)2 (10)
3

x
0.625π
x = 746.1282552 K
Whole trees needed (n)
n
50, 000
(4.75)(746.1282552)
n = 14.10791739
V = 0.625π
15 whole trees
Rubric:
[4]
15, and correct work is shown.
[3]
Appropriate work is shown, but one computational error is made.
or
Appropriate work is shown, but 15 is not identified as the answer.
[3]
[3]
Appropriate work is shown to find the volume and weight of one tree and amount of
money for the sale of one tree. No further correct work is shown.
[2] Appropriate work is shown, but two or more computational or rounding errors are
made.
or
[2] Appropriate work is shown, but one conceptual error is made.
or
[2] Appropriate work is shown to find the volume and weight of one tree, but no further
correct work is shown.
[1]
[1]
[1]
[0]
Appropriate work is shown, but one conceptual and one computational or rounding
error are made.
or
Appropriate work is shown to find the volume of one tree, but no further correct
work is shown.
or
15, but no work is shown.
A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
26
May 2014
Common Core Sample Question #13
13 In the diagram below, secant ACD and tangent AB are drawn from external point A
to circle O.
Prove the theorem: If a secant and a tangent are drawn to a circle from an external
point, the product of the lengths of the secant segment and its
external segment equals the length of the tangent segment
squared. ( AC  AD  AB 2 )
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May 2014
Key: See proof in the rationale below.
Measures CCLS: G-SRT.B, G-C.A
Mathematical Practice: 3, 6
Commentary: This question measures G-SRT.B because the student is required to use
similarity criteria to prove relationships in a geometric figure. It also
aligns to G-C.A because the student would use angles formed by chords
to prove the triangle are similar.
Rationale:
Statements
Reasons
1. Circle O, Secant ACD,
1. Given
Tangent AB
2. Chords BC and BD are drawn
2. Auxiliary lines
  BC

3. A  A, BC
3. Reflexive property

4. mBDC  m BC
4. The measure of an inscribed angle is half the
measure of the intercepted arc.
5. The measure of an angle formed by a tangent
and a chord is half the measure of the
intercepted arc.
6. BDC  CBA
6. Angles equal to half of the same arc are
congruent.
1
2
1 
5. mCBA  m BC
2
7.
8.
ABC  ADB
AB
AC

AD
AB
9. AC  AD  AB 2
7. AA
8. Corresponding sides of similar triangles are
proportional.
9. In a proportion, the product of the means
equals the product of the extremes.
Rubric:
[6]
A complete and correct proof that includes a concluding statement is written.
[5]
A proof is written that demonstrates a thorough understanding of the method of
proof and contains no conceptual errors, but one statement and/or reason is missing
or is incorrect.
28
May 2014
[4]
[4]
A proof is written that demonstrates a good understanding of the method of proof
and contains no conceptual errors, but two statements and/or reasons are missing or
are incorrect.
or
ABC  ADB, but no further correct work is shown.
[3]
A proof is written that demonstrates a good understanding of the method of proof,
but one conceptual error is made.
[2]
A proof is written that demonstrates a method of proof, but one conceptual error is
made, and one statement and/or reason is missing or is incorrect.
or
Some correct relevant statements about the proof are made, but three or four
statements or reasons are missing or are incorrect.
[2]
[1]
Only one correct relevant statement and reason are written.
[0]
The “given” and/or the “prove” statements are rewritten in the style of a formal
proof, but no further correct relevant statements are written.
or
A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
[0]
29
May 2014
Common Core Sample Question #14

14 Given: D is the image of A after a reflection over CH .

CH is the perpendicular bisector of BCE
ABC and DEC are drawn
Prove:
ABC  DEC
30
May 2014
Key: See proof in the rationale below.
Measures CCLS: G-CO.B
Mathematical Practice: 3, 6
Commentary: This question measures G-CO.B because the student is required to prove
that two triangles are congruent using the definition of congruence in
terms of rigid motion.
Rationale:
It is given that point D is the image of point A after a reflection in line CH.

It is given that CH is the perpendicular bisector of BE at point C. Since a bisector
divides a segment into two congruent segments at its midpoint, BC  EC . Point E is
the image of point B after a reflection over the line CH, since points B and E are

equidistant from point C and it is given that CH is perpendicular to BE.


Point C is on CH therefore, point C maps to itself after the reflection over CH .
Since all three vertices of triangle ABC map to all three vertices of triangle DEC
under the same line reflection, then ABC  DEC because a line reflection is a
rigid motion and triangles are congruent when one can be mapped onto the other
using a sequence of rigid motions.
Rubric:
[6]
A complete and correct proof that includes a concluding statement is written.
[5]
A proof is written that demonstrates a thorough understanding of the method of
proof and contains no conceptual errors, but one supporting statement and/or reason
is missing or is incorrect.
[4]
A proof is written that demonstrates a good understanding of the method of proof
and contains no conceptual errors, but two supporting statements and/or reasons are
missing or are incorrect.
[3]
A proof is written that demonstrates a good understanding of the method of proof,
but one conceptual error is made.
[2]
A proof is written that demonstrates a method of proof, but one conceptual error is
made, and one supporting statement and/or reason is missing or is incorrect.
[1]
Only one correct relevant statement and reason are written.
31
May 2014
[0]
[0]
The “given” and/or the “prove” statements are rewritten, but no further correct
relevant statements are written.
or
A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
32
May 2014